In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x^3−x−1; between 1 and 2

The Intermediate Value Theorem states that if a continuous function takes on two values at two points, then it must take on every value between those two points at least once. This theorem is crucial for proving the existence of roots in a given interval, as it guarantees that if the function changes signs over that interval, a real zero exists.

A continuous function is one where small changes in the input result in small changes in the output, meaning there are no breaks, jumps, or holes in the graph. For the Intermediate Value Theorem to apply, the function in question must be continuous over the interval being considered, which is true for all polynomial functions.

Polynomials are mathematical expressions involving variables raised to whole number powers, combined using addition, subtraction, and multiplication. They are continuous and differentiable everywhere on the real number line, which makes them suitable for applying the Intermediate Value Theorem to find real zeros within specified intervals.